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$$abc$$ allows us to count squarefrees. (English) Zbl 0924.11018
For given any polynomial $$f(x)\in \mathbb{Z}[x]$$, let $$B$$ be the greatest common divisor of $$f(n)$$ for all integers $$n$$, and let $$B'$$ be the smallest divisor of $$B$$ such that $$B/B'$$ is squarefree. Then $$f(n)/B'$$ can feasibly be squarefree for various integers $$n$$. It was P. Erdős [J. Lond. Math. Soc. 28, 416-425 (1953; Zbl 0051.27703)] who showed that if $$f(x)$$ has degree $$\leq 3$$ and $$B=1$$, then there are infinitely many integers $$n$$ for which $$f(n)$$ is squarefree.
In this paper the author assumes the well-known $$abc$$-conjecture to investigate this problem together with some others in related topics. He proves that under the $$abc$$-conjecture if $$f(x)$$ has no repeated roots then there are $$\sim c_fN$$ positive integers $$n\leq N$$ for which $$f(n)/ B'$$ is squarefree where $$c_f>0$$ is a constant. An explicit formula to determine $$c_f$$ is provided in the paper. Similar results are obtained for homogeneous $$f(x,y)\in \mathbb{Z}[x,y]$$ without any repeated linear factors.
Related topics are also discussed such as lower bounds for the products $$\prod_{p\mid f(m,n)}p$$ and $$\prod_{p\mid g(m)}p$$ where $$f(x,y)\in \mathbb{Z}[x,y]$$ and $$g(x)\in \mathbb{Z}[x]$$; and upper bounds for $$s_{n+1}-s_n$$ and an asymptotic expression for its average moments where $$s_n$$ is the increasing sequence of squarefree positive integers. These give more general or better results than those by several authors, e.g. N. Elkies [Int. Math. Res. Not. 1991, 99-109 (1991; Zbl 0763.11016)], M. Langevin [Sémin. Théorie des Nombres 1993-94, Publ. Math. Univ. Caen], M. Filaseta and O. Trifonov [Proc. Lond. Math. Soc. (3) 73, 241-278 (1996; Zbl 0867.11053)].

##### MSC:
 11C08 Polynomials in number theory 11N25 Distribution of integers with specified multiplicative constraints
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